94.09 Lattice polygons and the number 12: an elementaryproof

摘要

Among the most famous numbers a place can be reserved for thenumber 12. Some manifestations of 12 are listed in [1, section 3], and a fewothers are given at the end of the present article. The aim of this paper is toinvestigate (using only elementary linear algebra) the main aspects of one ofthese problems, which involves the number 12 and plane geometry. Themain protagonists of our story are lattice polygons, that is polygons whosevertices are points with integer coordinates (lattice points). For example, 9'1and P2 in Figure 1 are lattice polygons. Moreover 9'1 is convex (i.e. the linesegment joining any two points in 3'1 lies entirely in 9'1), but 42 is not.Besides 42 is the dual polygon of . This means that, if ph ... .1), are thelattice points on the boundary of 9'i arranged counterclockwise, then 42 canbe drawn by connecting the points q, = p, +1 — pi by segments in the orderof the indices. (From now on, the indices are taken modulo n so thatPn + 1 = pi.) Observe that the definition does not depend on the chosencoordinate system. We will denote by 9'v the dual of a generic latticepolygon 9'. Sometimes the dual of a convex polygon is not convex, as inFigure 1. This is still true if we reduce slightly the number of vertices. Forexample, can you find an example of a convex lattice polygon 9", with threeinternal lattice points and fewer than five vertices, whose dual 9'" is notconvex? A solution is given in section 3.